[Paper Review] On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor

Jan van den Heuvel, Patrice Ossona de Mendez+3 more|arXiv (Cornell University)|Feb 29, 2016

Advanced Graph Theory Research25 references44 citations

TL;DR

This paper establishes tight linear and polynomial upper bounds for the generalized coloring numbers colr(G) and wcolr(G) in graphs excluding a fixed minor, significantly improving prior exponential bounds. By leveraging lexicographic breadth-first search trees and structural graph properties in minor-closed families, the authors prove colr(G) ≤ (t−1)/2 · (2r + 1) and wcolr(G) ≤ O(rt−1) for Kt-minor-free graphs, with tighter bounds for planar and bounded-genus graphs.

ABSTRACT

The generalised colouring numbers $\mathrm{col}_r(G)$ and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the $r$-colouring number $\mathrm{col}_r$ and a polynomial bound for the weak $r$-colouring number $\mathrm{wcol}_r$. In particular, we show that if $G$ excludes $K_t$ as a minor, for some fixed $t\ge4$, then $\mathrm{col}_r(G)\le\binom{t-1}{2}\,(2r+1)$ and $\mathrm{wcol}_r(G)\le\binom{r+t-2}{t-2}\cdot(t-3)(2r+1)\in\mathcal{O}(r^{\,t-1})$. In the case of graphs $G$ of bounded genus $g$, we improve the bounds to $\mathrm{col}_r(G)\le(2g+3)(2r+1)$ (and even $\mathrm{col}_r(G)\le5r+1$ if $g=0$, i.e. if $G$ is planar) and $\mathrm{wcol}_r(G)\le\Bigl(2g+\binom{r+2}{2}\Bigr)\,(2r+1)$.

Motivation & Objective

• To improve upon the exponential upper bounds for generalized coloring numbers in graphs excluding a fixed minor, which were previously established by Grohe et al.
• To provide tight, explicit bounds for colr(G) and wcolr(G) in Kt-minor-free graphs, particularly for planar and bounded-genus graphs.
• To establish connections between generalized coloring numbers and structural graph parameters such as tree-depth, tree-width, and genus.
• To demonstrate that these bounds yield improved results for related graph invariants, such as the acyclic chromatic number.
• To refine existing bounds in sparse graph classes by exploiting lexicographic breadth-first search orderings and minor-exclusion properties.

Proposed method

• Utilizes lexicographic breadth-first search (LexBFS) trees to define a vertex ordering that controls reachability within radius r.
• Applies structural graph theory to bound the number of vertices that can be strongly r-reachable from any vertex u under a LexBFS order.
• Employs path separation arguments using parent paths (Pa, Pb, Pc) in a maximal planar graph to constrain the size of r-neighborhoods.
• Leverages the fact that vertices found earlier in LexBFS have lower order, enabling inductive control over reachability sets.
• Uses induction and level-wise analysis in the BFS tree to bound the number of vertices in NGr[u] ∩ V(Pa), V(Pb), and V(Pu).
• Derives bounds via case analysis on the face containing u’s parent, especially distinguishing between internal and outer faces in planar graphs.

Experimental results

Research questions

• RQ1 What are the tightest possible upper bounds for the r-coloring number colr(G) in graphs excluding a fixed minor?
• RQ2 How do these bounds depend on the minor excluded, particularly for complete graphs Kt?
• RQ3 Can the weak r-coloring number wcolr(G) be bounded polynomially in r for Kt-minor-free graphs?
• RQ4 How do the bounds improve in special graph classes such as planar or bounded-genus graphs?
• RQ5 To what extent do these bounds improve known upper bounds for related invariants like the acyclic chromatic number?

Key findings

• For any Kt-minor-free graph G with t ≥ 4, colr(G) ≤ (t−1)/2 · (2r + 1), a linear bound in r.
• For the same class, wcolr(G) ≤ (r+t−2 choose t−2) · (t−3)(2r + 1), which is O(rt−1), a polynomial bound in r.
• In planar graphs (genus g = 0), colr(G) ≤ 5r + 1, and this bound is tight for r = 1.
• For graphs of genus g, colr(G) ≤ (4g + 5)r + 2g + 1, improving upon previous exponential bounds.
• For planar graphs, wcolr(G) ≤ (r+2 choose 2) · (2r + 1), which is O(r³), and this bound is also tight for r = 1.
• The acyclic chromatic number of Kt-minor-free graphs is now bounded by O(t²), improving upon the prior O(t² log²t) bound.